# Integral domain

2020 Mathematics Subject Classification: *Primary:* 13G05 [MSN][ZBL]

*integral ring*

A commutative ring with identity and without divisors of zero (cf. Zero divisor). Any field, and also any ring with identity contained in a field, is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the field of fractions.

If $A$ is an integral domain, then the ring of polynomials $A[X]$ and the ring of formal power series $A[[X]]$ over $A$ are also integral domains. If $A$ is a commutative ring with identity and $I$ is any ideal in $A$, then the ring $A/I$ is an integral domain if and only if $I$ is a prime ideal. A ring $A$ without nilpotents is an integral domain if and only if the spectrum of $A$ is an irreducible topological space (cf. Spectrum of a ring).

Sometimes commutativity of $A$ is not required in the definition of an integral domain. Skew-fields and subrings of a skew-field containing the identity are examples of non-commutative integral domains. However, it is not true, in general, that an arbitrary non-commutative integral domain can be imbedded in a skew-field (see [2], and Imbedding of rings).

#### References

[1] | S. Lang, "Algebra" , Addison-Wesley (1965) |

[2] | P.M. Cohn, "Free rings and their relations" , Acad. Press (1985) |

[3] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) |

**How to Cite This Entry:**

Integral domain.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Integral_domain&oldid=39096